OX THE CORRESPONDENCE OF TWO POINTS ON A CURVE. 
12 
[385 
tangents from P' to the curve, that is, a. = n — 2. Hence the number of inflexions is 
= (m — 2) + (n — 2) + 4D, = m + n — 4 + 2 (w — 2in + 2), = 3 (n — m), which is right. 
8. For the purpose of the next example it is necessary to present the fundamental 
equation a = a + a' + 2kD under a more general form. The curve © may intersect the 
given curve in a system of points P' each p times, a system of points Q' each 
q times, &c., in such manner that the points (P, P'), the points (P, Q'), &c., are pairs 
of points corresponding to each other according to distinct laws; and we shall then 
have the numbers (a, a, a), (b, /3, /3'), &c., belonging to these pairs respectively; viz. 
(P, P') are points having an (a, a') correspondence, and the number of united points 
is = a; similarly (P, Q') are points having a (/3, ¡3') correspondence, and the number 
of united points is = b; and so on. The theorem then is 
p (a — a — a) 4- q (b — /3 — /3') 4- ... = 2 kD. 
9. Investigation of the number of double tangents:—Take P', an intersection with 
the curve of a tangent drawn from P to the curve (or what is the same thing, P, P' 
cotangentials of any point of the curve); the united points are here the points of 
contact of the several double tangents of the curve; or if t be the number of double 
tangents, then the number of united points is = 2t. The curve © is the system of 
the n — 2 tangents from P to the curve; each tangent has with the curve 1 inter 
section at P, that is, Jc = n— 2; each tangent, besides, meets the curve in the point 
of contact Q' twice, and in (m — 3) points P'. Hence, if (a, a, a') refer to the points 
(P, Q'), and (2t, /3, /3') to the points (P, P'), we have 
2 (a — a — a!) + 2t — /3 — /3' = 2 (n — 2) D. 
Moreover, from the last example the value of a — a — a! is = 4P, and the formula thus 
becomes 
2r — /3 — /3' = 2 (n — 6) D; 
but from above it appears that we have ¡3 = ¡3' = (n — 2) (m — 3), whence 
2t = 2 (n — 2) (in — 3) + 2 (n — G) D, 
= 2 (n— 2) (m — 3) + (n — 6) (n — 2m + 2), 
= n 2 — 10?i + 8 m, 
which is right; in fact, observing that t (the number of inflexions) is = 3n — 3in, the 
formula is equivalent to 2t + 3t = n- — n — in, that is, m = n 2 - n — 2r — Si. 
In the foregoing examples the curve © is a line or system of lines; but I give 
an example in which © is a system of conics, and in which, as will appear, we have 
to consider the two characteristics (/a, v) of the system. 
10. Investigation of the number of conics which can be drawn, satisfying any 
four conditions, and touching the given curve; or say of the number of the conics 
(4Z) (1). Take P', an intersection of the given curve by a conic (4Z) passing through 
the point P, then the number of the united points is equal to that of the conics 
(4Z) (1). The curve © is here the system of the conics (4Z) which pass through P;